> On 20 Jun., 15:41, "Julio Di Egidio"> > So, there is no such thing as an "additional step" to take, we> > just consider the limit of the sequence...> > No. You seem to have not yet understood. But I enjoy to explain it> again, and again, and again, ... because the story is incredible.> > Every set of the sequence> > 1,> 1, 2> 1, 2, 3> ...> > is the union of all its predecessors and its last {n}.> > There are infinitely many sets, so there are infinitely many unions.> > None of these infinitely many unions yields |N

The reason being that each such union os of only finitely many FISONs all of which are necessarily contained in a largest FISON of that unioning.

The union of a non-empty set of FISONs will result in |N if and only if that set does NOT contain any maximal FISON. Which condition requires that the set of FISONs being unioned be an infinite set of FISONs.

Thus the union of any infinite set of FISONs equals |N but the union of any finite non-empty set of FISONs is the maximal FISON in that set.

This is so simple and straightforwrd that it is amazing, at least to to everyone but WM, that WM is incapable of grasping it.

> After you pause, exhausted, since you failed to get |N but got only> sets which a lacking aleph_0 natural numbers

Only WM is ever in such a punishing predicament.For the rest of us, we merely ask whether a given non-empty set of FISONs has a largest member.

If so the union is that largest member.

If not, the union is |N.

So outside of WMytheology such unions are trivial, and it is only inside WMytheology that anyone need have any problems.--